Question

Calcule a integral indefinida

t) $\int\ {} (\frac{tgx}{secx}) \, dx$

Answer 1

Sabendo que

tgx=senx/cosx
secx=1/cosx

A integral indefinida

$\int\limits { \frac{tgx}{secx} } \, dx = \int\limits { \frac{ \frac{senx}{cosx} }{ \frac{1}{cosx} } } \, dx$

$\int\limits{ \frac{senxcosx}{cosx} } \, dx =$

$\int\limits {senx} \, dx$

$-cos x + C$

Answer 2

$\sf \displaystyle \int \left(\frac{tg\:x}{sec\:x}\right)dx\\\\\\=\int \sin \left(x\right)dx\\\\\\=-\cos \left(x\right)\\\\\\\to \boxed{\sf =-\cos \left(x\right)+C}$